from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

__a    :	Optional[int]         =    TypeVar("""T""")





def 							UpperCAmelCase		(       lowercase				):







				"""simple docstring"""



				return (position - 1) // 2





def 							UpperCAmelCase		(       lowercase				):







				"""simple docstring"""



				return (2 * position) + 1





def 							UpperCAmelCase		(       lowercase				):







				"""simple docstring"""



				return (2 * position) + 2




class   _UpperCamelCase (		Generic[T]					):


			"""simple docstring"""






			def __init__(  self		)	->   None:

							'''simple docstring'''


							__lowercase					     =				[]
							__lowercase					     =				{}
							__lowercase					     =				0





			def __len__(  self		)	->   int:

							'''simple docstring'''


							return self.elements





			def __repr__(  self		)	->   str:

							'''simple docstring'''


							return str(self.heap		)





			def 			_SCREAMING_SNAKE_CASE     (  self		)	->   bool:

							'''simple docstring'''


							return self.elements == 0





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__ ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							self.heap.append((elem, weight)		)
							__lowercase					     =				self.elements
							self.elements += 1
							self._bubble_up(lowerCAmelCase__		)





			def 			_SCREAMING_SNAKE_CASE     (  self		)	->   T:

							'''simple docstring'''


							if self.elements > 1:
											self._swap_nodes(0 ,  self.elements - 1		)
							__lowercase     ,	__lowercase					     =				self.heap.pop()
							del self.position_map[elem]
							self.elements -= 1
							if self.elements > 0:
											__lowercase     ,	__lowercase					     =				self.heap[0]
											self._bubble_down(lowerCAmelCase__		)
							return elem





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__ ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							__lowercase					     =				self.position_map[elem]
							__lowercase					     =				(elem, weight)
							if position > 0:
											__lowercase					     =				get_parent_position(lowerCAmelCase__		)
											__lowercase     ,	__lowercase					     =				self.heap[parent_position]
											if parent_weight > weight:
															self._bubble_up(lowerCAmelCase__		)
											else:
															self._bubble_down(lowerCAmelCase__		)
							else:
											self._bubble_down(lowerCAmelCase__		)





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							__lowercase					     =				self.position_map[elem]
							if curr_pos == 0:
											return None
							__lowercase					     =				get_parent_position(lowerCAmelCase__		)
							__lowercase     ,	__lowercase					     =				self.heap[curr_pos]
							__lowercase     ,	__lowercase					     =				self.heap[parent_position]
							if parent_weight > weight:
											self._swap_nodes(lowerCAmelCase__ ,  lowerCAmelCase__		)
											return self._bubble_up(lowerCAmelCase__		)
							return None





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							__lowercase					     =				self.position_map[elem]
							__lowercase     ,	__lowercase					     =				self.heap[curr_pos]
							__lowercase					     =				get_child_left_position(lowerCAmelCase__		)
							__lowercase					     =				get_child_right_position(lowerCAmelCase__		)
							if child_left_position < self.elements and child_right_position < self.elements:
											__lowercase     ,	__lowercase					     =				self.heap[child_left_position]
											__lowercase     ,	__lowercase					     =				self.heap[child_right_position]
											if child_right_weight < child_left_weight and child_right_weight < weight:
															self._swap_nodes(lowerCAmelCase__ ,  lowerCAmelCase__		)
															return self._bubble_down(lowerCAmelCase__		)
							if child_left_position < self.elements:
											__lowercase     ,	__lowercase					     =				self.heap[child_left_position]
											if child_left_weight < weight:
															self._swap_nodes(lowerCAmelCase__ ,  lowerCAmelCase__		)
															return self._bubble_down(lowerCAmelCase__		)
							else:
											return None
							if child_right_position < self.elements:
											__lowercase     ,	__lowercase					     =				self.heap[child_right_position]
											if child_right_weight < weight:
															self._swap_nodes(lowerCAmelCase__ ,  lowerCAmelCase__		)
															return self._bubble_down(lowerCAmelCase__		)
							return None





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__ ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							__lowercase					     =				self.heap[nodea_pos][0]
							__lowercase					     =				self.heap[nodea_pos][0]
							__lowercase     ,	__lowercase					     =				(
							    self.heap[nodea_pos],
							    self.heap[nodea_pos],
							)
							__lowercase					     =				nodea_pos
							__lowercase					     =				nodea_pos




class   _UpperCamelCase (		Generic[T]					):


			"""simple docstring"""






			def __init__(  self		)	->   None:

							'''simple docstring'''


							__lowercase					     =				{}
							__lowercase					     =				0





			def __repr__(  self		)	->   str:

							'''simple docstring'''


							return str(self.connections		)





			def __len__(  self		)	->   int:

							'''simple docstring'''


							return self.nodes





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							if node not in self.connections:
											__lowercase					     =				{}
											self.nodes += 1





			def 			_SCREAMING_SNAKE_CASE     (  self ,  lowerCAmelCase__ ,  lowerCAmelCase__ ,  lowerCAmelCase__		)	->   None:

							'''simple docstring'''


							self.add_node(lowerCAmelCase__		)
							self.add_node(lowerCAmelCase__		)
							__lowercase					     =				weight
							__lowercase					     =				weight





def 							UpperCAmelCase		(       lowercase				,     ):







				"""simple docstring"""



				__lowercase					     =				{node: maxsize for node in graph.connections}
				__lowercase					     =				{node: None for node in graph.connections}

				__lowercase					     =				MinPriorityQueue()
				for node, weight in dist.items():
								priority_queue.push(lowercase				,     lowercase				)

				if priority_queue.is_empty():
								return dist, parent

				# initialization
				__lowercase					     =				priority_queue.extract_min()
				__lowercase					     =				0
				for neighbour in graph.connections[node]:
								if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
												__lowercase					     =				dist[node] + graph.connections[node][neighbour]
												priority_queue.update_key(lowercase				,     dist[neighbour]				)
												__lowercase					     =				node

    # running prim's algorithm
				while not priority_queue.is_empty():
								__lowercase					     =				priority_queue.extract_min()
								for neighbour in graph.connections[node]:
												if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
																__lowercase					     =				dist[node] + graph.connections[node][neighbour]
																priority_queue.update_key(lowercase				,     dist[neighbour]				)
																__lowercase					     =				node
				return dist, parent
def 							UpperCAmelCase		(       lowercase				,     lowercase				):







				"""simple docstring"""



				__lowercase					     =				word.split()

				def justify(lowercase				,     lowercase				,     lowercase				) -> str:
								__lowercase					     =				max_width - width
								__lowercase					     =				len(lowercase				)
								if len(lowercase				) == 1:
												# if there is only word in line
												# just insert overall_spaces_count for the remainder of line
												return line[0] + " " * overall_spaces_count
								else:
												__lowercase					     =				words_count - 1
												# num_spaces_between_words_list[i] : tells you to insert
												# num_spaces_between_words_list[i] spaces
												# after word on line[i]
												__lowercase					     =				spaces_to_insert_between_words * [
												    overall_spaces_count // spaces_to_insert_between_words
												]
												__lowercase					     =				(
												    overall_spaces_count % spaces_to_insert_between_words
												)
												# distribute spaces via round robin to the left words
												for i in range(lowercase				):
																num_spaces_between_words_list[i] += 1
												__lowercase					     =				[]
												for i in range(lowercase				):
																# add the word
																aligned_words_list.append(line[i]				)
																# add the spaces to insert
																aligned_words_list.append(num_spaces_between_words_list[i] * ''' '''				)
												# just add the last word to the sentence
												aligned_words_list.append(line[-1]				)
												# join the aligned words list to form a justified line
												return "".join(lowercase				)

				__lowercase					     =				[]
				__lowercase					     =				[]
				__lowercase					     =				0
				for word in words:
								if width + len(lowercase				) + len(lowercase				) <= max_width:
												# keep adding words until we can fill out max_width
												# width = sum of length of all words (without overall_spaces_count)
												# len(word) = length of current word
												# len(line) = number of overall_spaces_count to insert between words
												line.append(lowercase				)
												width += len(lowercase				)
								else:
												# justify the line and add it to result
												answer.append(justify(lowercase				,     lowercase				,     lowercase				)				)
												# reset new line and new width
												__lowercase     ,	__lowercase					     =				[word], len(lowercase				)
				__lowercase					     =				max_width - width - len(lowercase				)
				answer.append(''' '''.join(lowercase				) + (remaining_spaces + 1) * ''' '''				)
				return answer


if __name__ == "__main__":
		from doctest import testmod

		testmod()